Iterated Operations and Operations over Finite Sets Take Values in the Underlying Set

In the setting of Commutative Rings, Fields and Ordered Fields: Standard Notation, let $\ast$ be a binary operation on a set $X$, and let $[n]$ be as in Intervals of Natural Numbers §segment for $n\in\mathbb{N}$.

¶ For every $n\in\mathbb{N}$ and every map $a:[n]\to X$, $k\mapsto a_{k}$, the iterated operation $\mathop{\ast}\limits_{k=1}^{n}a_{k}$ lies in $X$.

¶ Let $\ast$ be associative and commutative, let $A$ be a finite set and $f:A\to X$ a map, and let $A\neq\emptyset$ or let $\ast$ have a neutral element $e$. Then the iterated operation $\mathop{\ast}\limits_{x\in A}f(x)$ over $A$, as in Sums and Products over a Finite Set and over an Interval §operation and Sums and Products over a Finite Set and over an Interval §empty, lies in $X$.

In the clauses finite-sums and segment-sums, let $R$, with $+$, $\cdot$, $0$ and $1$, be a commutative ring; finite sums and products of elements of $R$ are as in Commutative Rings, Fields and Ordered Fields: Standard Notation §rings, and along $[n]$ as in Iterated Operations: Finite Sums and Finite Products §sums.

¶ For every finite set $A$ and every map $f:A\to R$, $\sum_{x\in A}f(x)\in R$ and $\prod_{x\in A}f(x)\in R$.

¶ For every $n\in\mathbb{N}$ and every map $a:[n]\to R$, $\sum_{k=1}^{n}a_{k}\in R$ and $\prod_{k=1}^{n}a_{k}\in R$.